In
mathematics, a character is (most commonly) a special kind of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
to a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
(such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s). There are at least two distinct, but overlapping meanings.
Other uses of the word "character" are almost always qualified.
Multiplicative character
A multiplicative character (or linear character, or simply character) on a group ''G'' is a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
from ''G'' to the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of a field , usually the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under pointwise multiplication.
This group is referred to as the
character group In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in t ...
of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
); other such homomorphisms are then called ''quasi-characters''.
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
s can be seen as a special case of this definition.
Multiplicative characters are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, i.e. if
are different characters on a group ''G'' then from
it follows that
.
Character of a representation
The character
of a
representation of a group ''G'' on a
finite-dimensional vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' over a field ''F'' is the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the representation
, i.e.
:
for
In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "
character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
" and one-dimensional characters are also called "linear characters" within this context.
Alternative definition
If restricted to
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
abelian group with
representation in
(i.e.
), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a
direct sum of
representations. For non-abelian groups, the original definition would be more general than this one):
A character
of group
is a group homomorphism
i.e.
for all
If
is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by
where
is the
circle group.
See also
*
Character group In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in t ...
*
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
*
Harish-Chandra character In mathematics, the Harish-Chandra character, named after Harish-Chandra, of a representation of a semisimple Lie group ''G'' on a Hilbert space ''H'' is a distribution on the group ''G'' that is analogous to the character of a finite-dimensional ...
*
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
*
Infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group ''G'' on a vector space ''V'' is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagona ...
*
Alternating character
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total o ...
*
Characterization (mathematics)
In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X ...
*
Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
References
* Lectures Delivered at the University of Notre Dame
*
External links
* {{springer, title=Character of a group, id=p/c021560
Representation theory